| L(s) = 1 | − 3.64·5-s + 7-s − 2·11-s − 4·13-s + 0.354·17-s + 19-s + 6·23-s + 8.29·25-s − 5.64·29-s + 1.29·31-s − 3.64·35-s + 5.29·37-s + 8.58·41-s − 2·43-s − 2.35·47-s + 49-s + 8.93·53-s + 7.29·55-s − 14.5·59-s + 2.70·61-s + 14.5·65-s + 14.5·67-s − 0.354·71-s − 9.29·73-s − 2·77-s + 11.2·79-s − 16.9·83-s + ⋯ |
| L(s) = 1 | − 1.63·5-s + 0.377·7-s − 0.603·11-s − 1.10·13-s + 0.0859·17-s + 0.229·19-s + 1.25·23-s + 1.65·25-s − 1.04·29-s + 0.231·31-s − 0.616·35-s + 0.869·37-s + 1.34·41-s − 0.304·43-s − 0.343·47-s + 0.142·49-s + 1.22·53-s + 0.983·55-s − 1.89·59-s + 0.346·61-s + 1.80·65-s + 1.78·67-s − 0.0420·71-s − 1.08·73-s − 0.227·77-s + 1.27·79-s − 1.85·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.64T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 0.354T + 17T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 5.64T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 - 8.58T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 - 8.93T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 0.354T + 71T^{2} \) |
| 73 | \( 1 + 9.29T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41273172455490866247135357414, −7.04173865384601916746609387992, −5.92219760382809213557430117564, −5.05286515177346034143044927329, −4.61321328386928231412970215372, −3.86136669335360337338834127199, −3.07352470371142857286568009715, −2.35443053532807852030180187365, −0.983061595614113974541227485583, 0,
0.983061595614113974541227485583, 2.35443053532807852030180187365, 3.07352470371142857286568009715, 3.86136669335360337338834127199, 4.61321328386928231412970215372, 5.05286515177346034143044927329, 5.92219760382809213557430117564, 7.04173865384601916746609387992, 7.41273172455490866247135357414
